-
Previous Article
Non-autonomous weakly damped plate model on time-dependent domains
- DCDS-S Home
- This Issue
-
Next Article
Existence and multiplicity of positive solutions for a class of quasilinear Schrödinger equations in $ \mathbb R^N $$ ^\diamondsuit $
$ W^{2, p} $-regularity for asymptotically regular fully nonlinear elliptic and parabolic equations with oblique boundary values
1. | School of Mathematical Sciences, Hebei Normal University, Shijiazhuang, Hebei 050016, China |
2. | Department of Mathematics, Beijing Jiaotong University, Beijing 100044, China |
3. | School of Sciences, Hebei University of Science and Technology, Shijiazhuang, Hebei 050016, China |
We prove a global $ W^{2, p} $-estimate for the viscosity solution to fully nonlinear elliptic equations $ F(x, u, Du, D^{2}u) = f(x) $ with oblique boundary condition in a bounded $ C^{2, \alpha} $-domain for every $ \alpha\in (0, 1) $. Here, the nonlinearities $ F $ is assumed to be asymptotically $ \delta $-regular to an operator $ G $ that is $ (\delta, R) $-vanishing with respect to $ x $. We employ the approach of constructing a regular problem by an appropriate transformation. With a similar argument, we also obtain a global $ W^{2, p} $-estimate for the viscosity solution to fully nonlinear parabolic equations $ F(x, t, u, Du, D^{2}u)-u_{t} = f(x, t) $ with oblique boundary condition in a bounded $ C^{3} $-domain.
References:
[1] |
T. Alberico, C. Capozzoli, R. Schiattarella and L. D'Onofrio,
G-convergence for non-divergence elliptic operators with VMO coefficients in $\mathbb{R}^3$, Discrete Contin. Dyn. Syst. Ser. S, 12 (2019), 129-137.
doi: 10.3934/dcdss.2019009. |
[2] |
S.-S. Byun and J. Han,
$W^{2, p}$-estimates for fully nonlinear elliptic equations with oblique boundary conditions, J. Differential Equations, 268 (2020), 2125-2150.
doi: 10.1016/j.jde.2019.09.018. |
[3] |
S.-S. Byun and J. Han, $L^{p}$-estimates for the Hessians of solutions to fully nonlinear parabolic equations with oblique boundary conditions, arXiv: 2012.07435v1 [math.AP], 43 pages. |
[4] |
S.-S. Byun, M. Lee and D. K. Palagachev,
Hessian estimates in weighted Lebesgue spaces for fully nonlinear elliptic equations, J. Differential Equations, 260 (2016), 4550-4571.
doi: 10.1016/j.jde.2015.11.025. |
[5] |
S.-S. Byun, J. Oh and L. Wang,
Global Calderón-Zygmund theory for asymptotically regular nonlinear elliptic and parabolic equations, Int. Math. Res. Not. IMRN, 2015 (2015), 8289-8308.
doi: 10.1093/imrn/rnu203. |
[6] |
L. A. Caffarelli,
Interior a priori estimates for solutions of fully nonlinear equations, Ann. of Math., 130 (1989), 189-213.
doi: 10.2307/1971480. |
[7] |
L. A. Caffarelli and Q. Huang,
Estimates in the generalized Campamato-John-Nirenberg spaces for fully nonlinear elliptic equations, Duke Math. J., 118 (2003), 1-17.
doi: 10.1215/S0012-7094-03-11811-6. |
[8] |
M. Chipot and L. C. Evans,
Linearisation at infinity and Lipschitz estimates for certain problems in the calculus of variations, Proc. Roy. Soc. Edinburgh Sect. A, 102 (1986), 291-303.
doi: 10.1017/S0308210500026378. |
[9] |
J. Choi, H. Dong and D. Kim,
Conormal derivative problems for stationary Stokes system in Sobolev spaces, Discrete Contin. Dyn. Syst., 38 (2018), 2349-2374.
doi: 10.3934/dcds.2018097. |
[10] |
H. Dong and N. V. Krylov,
On the existence of smooth solutions for fully nonlinear parabolic equations with measurable "coefficients" without convexity assumptions, Commun. Partial Diff. Equ., 38 (2013), 1038-1068.
doi: 10.1080/03605302.2012.756013. |
[11] |
H. Dong, N. V. Krylov and X. Li,
On fully nonlinear elliptic and parabolic equations with VMO coefficients in domains, St. Petersburg Math. J., 24 (2013), 39-69.
doi: 10.1090/S1061-0022-2012-01231-8. |
[12] |
L. Escauriaza,
$W^{2, n}$ a priori estimates for solutions to fully nonlinear equations, Indiana Univ. Math. J., 42 (1993), 413-423.
doi: 10.1512/iumj.1993.42.42019. |
[13] |
M. Foss,
Global regularity for almost minimizers of nonconvex variational problems, Ann. Mat. Pura Appl., 187 (2008), 263-321.
doi: 10.1007/s10231-007-0045-2. |
[14] |
C. Imbert and L. Silvestre,
$C^{1, \alpha}$ regularity of solutions of some degenerate fully nonlinear elliptic equations, Advances in Mathematics, 233 (2013), 196-206.
doi: 10.1016/j.aim.2012.07.033. |
[15] |
N. V. Krylov,
Linear and fully nonlinear elliptic equations with $L_{d}$-drift, Commun. Partial Diff. Equ., 45 (2020), 1778-1798.
doi: 10.1080/03605302.2020.1805462. |
[16] |
T. Kuusi and G. Mingione, New perturbation methods for nonlinear parabolic problems, J. Math. Pures Appl., 98 (2012), 390-427.
doi: 10.1016/j.matpur.2012.02.004. |
[17] |
D. Li and K. Zhang,
Regularity for fully nonlinear elliptic equations with oblique boundary conditions, Arch. Ration. Mech. Anal., 228 (2018), 923-967.
doi: 10.1007/s00205-017-1209-x. |
[18] |
S. Liang and S. Zheng, Calderón-Zygmund estimate for asymptotically regular non-uniformly elliptic equations, J. Math. Anal. Appl., 484 (2020), 123749, 17 pp.
doi: 10.1016/j.jmaa.2019.123749. |
[19] |
P. Marcellini,
Regularity under general and $p, q$-growth conditions, Discrete Contin. Dyn. Syst. Ser. S, 13 (2020), 2009-2031.
doi: 10.3934/dcdss.2020155. |
[20] |
E. Milakis and L. E. Silvestre,
Regularity for fully nonlinear elliptic equations with Neumann boundary data, Commun. Partial Diff. Equ., 31 (2006), 1227-1252.
doi: 10.1080/03605300600634999. |
[21] |
C. Scheven and T. Schmidt,
Asymptotically regular problems II: Partial Lipschitz continuity and a singular set of positive measure, Ann. Sc. Norm. Super. Pisa Cl. Sci., 8 (2009), 469-507.
doi: 10.2422/2036-2145.2009.3.04. |
[22] |
L. Wang,
On the regularity theory of fully nonlinear parabolic equations. I, Comm. Pure Appl. Math., 45 (1992), 27-76.
doi: 10.1002/cpa.3160450103. |
[23] |
N. Winter,
$W^{2, p}$ and $W^{1, p}$-estimates at the boundary for solutions of fully nonlinear, uniformly elliptic equations, Z. Anal. Anwend., 28 (2009), 129-164.
doi: 10.4171/ZAA/1377. |
[24] |
J. Zhang and S. Zheng,
Lorentz estimates for asymptotically regular fully nonlinear parabolic equations, Math. Nachr., 291 (2018), 996-1008.
doi: 10.1002/mana.201600497. |
[25] |
J. Zhang, M. Cai and S. Zheng,
Weighted Lorentz estimate for asymptotically regular parabolic equations of p(x, t)-Laplacian type, Nonlinear Anal., 180 (2019), 225-235.
doi: 10.1016/j.na.2018.10.013. |
show all references
References:
[1] |
T. Alberico, C. Capozzoli, R. Schiattarella and L. D'Onofrio,
G-convergence for non-divergence elliptic operators with VMO coefficients in $\mathbb{R}^3$, Discrete Contin. Dyn. Syst. Ser. S, 12 (2019), 129-137.
doi: 10.3934/dcdss.2019009. |
[2] |
S.-S. Byun and J. Han,
$W^{2, p}$-estimates for fully nonlinear elliptic equations with oblique boundary conditions, J. Differential Equations, 268 (2020), 2125-2150.
doi: 10.1016/j.jde.2019.09.018. |
[3] |
S.-S. Byun and J. Han, $L^{p}$-estimates for the Hessians of solutions to fully nonlinear parabolic equations with oblique boundary conditions, arXiv: 2012.07435v1 [math.AP], 43 pages. |
[4] |
S.-S. Byun, M. Lee and D. K. Palagachev,
Hessian estimates in weighted Lebesgue spaces for fully nonlinear elliptic equations, J. Differential Equations, 260 (2016), 4550-4571.
doi: 10.1016/j.jde.2015.11.025. |
[5] |
S.-S. Byun, J. Oh and L. Wang,
Global Calderón-Zygmund theory for asymptotically regular nonlinear elliptic and parabolic equations, Int. Math. Res. Not. IMRN, 2015 (2015), 8289-8308.
doi: 10.1093/imrn/rnu203. |
[6] |
L. A. Caffarelli,
Interior a priori estimates for solutions of fully nonlinear equations, Ann. of Math., 130 (1989), 189-213.
doi: 10.2307/1971480. |
[7] |
L. A. Caffarelli and Q. Huang,
Estimates in the generalized Campamato-John-Nirenberg spaces for fully nonlinear elliptic equations, Duke Math. J., 118 (2003), 1-17.
doi: 10.1215/S0012-7094-03-11811-6. |
[8] |
M. Chipot and L. C. Evans,
Linearisation at infinity and Lipschitz estimates for certain problems in the calculus of variations, Proc. Roy. Soc. Edinburgh Sect. A, 102 (1986), 291-303.
doi: 10.1017/S0308210500026378. |
[9] |
J. Choi, H. Dong and D. Kim,
Conormal derivative problems for stationary Stokes system in Sobolev spaces, Discrete Contin. Dyn. Syst., 38 (2018), 2349-2374.
doi: 10.3934/dcds.2018097. |
[10] |
H. Dong and N. V. Krylov,
On the existence of smooth solutions for fully nonlinear parabolic equations with measurable "coefficients" without convexity assumptions, Commun. Partial Diff. Equ., 38 (2013), 1038-1068.
doi: 10.1080/03605302.2012.756013. |
[11] |
H. Dong, N. V. Krylov and X. Li,
On fully nonlinear elliptic and parabolic equations with VMO coefficients in domains, St. Petersburg Math. J., 24 (2013), 39-69.
doi: 10.1090/S1061-0022-2012-01231-8. |
[12] |
L. Escauriaza,
$W^{2, n}$ a priori estimates for solutions to fully nonlinear equations, Indiana Univ. Math. J., 42 (1993), 413-423.
doi: 10.1512/iumj.1993.42.42019. |
[13] |
M. Foss,
Global regularity for almost minimizers of nonconvex variational problems, Ann. Mat. Pura Appl., 187 (2008), 263-321.
doi: 10.1007/s10231-007-0045-2. |
[14] |
C. Imbert and L. Silvestre,
$C^{1, \alpha}$ regularity of solutions of some degenerate fully nonlinear elliptic equations, Advances in Mathematics, 233 (2013), 196-206.
doi: 10.1016/j.aim.2012.07.033. |
[15] |
N. V. Krylov,
Linear and fully nonlinear elliptic equations with $L_{d}$-drift, Commun. Partial Diff. Equ., 45 (2020), 1778-1798.
doi: 10.1080/03605302.2020.1805462. |
[16] |
T. Kuusi and G. Mingione, New perturbation methods for nonlinear parabolic problems, J. Math. Pures Appl., 98 (2012), 390-427.
doi: 10.1016/j.matpur.2012.02.004. |
[17] |
D. Li and K. Zhang,
Regularity for fully nonlinear elliptic equations with oblique boundary conditions, Arch. Ration. Mech. Anal., 228 (2018), 923-967.
doi: 10.1007/s00205-017-1209-x. |
[18] |
S. Liang and S. Zheng, Calderón-Zygmund estimate for asymptotically regular non-uniformly elliptic equations, J. Math. Anal. Appl., 484 (2020), 123749, 17 pp.
doi: 10.1016/j.jmaa.2019.123749. |
[19] |
P. Marcellini,
Regularity under general and $p, q$-growth conditions, Discrete Contin. Dyn. Syst. Ser. S, 13 (2020), 2009-2031.
doi: 10.3934/dcdss.2020155. |
[20] |
E. Milakis and L. E. Silvestre,
Regularity for fully nonlinear elliptic equations with Neumann boundary data, Commun. Partial Diff. Equ., 31 (2006), 1227-1252.
doi: 10.1080/03605300600634999. |
[21] |
C. Scheven and T. Schmidt,
Asymptotically regular problems II: Partial Lipschitz continuity and a singular set of positive measure, Ann. Sc. Norm. Super. Pisa Cl. Sci., 8 (2009), 469-507.
doi: 10.2422/2036-2145.2009.3.04. |
[22] |
L. Wang,
On the regularity theory of fully nonlinear parabolic equations. I, Comm. Pure Appl. Math., 45 (1992), 27-76.
doi: 10.1002/cpa.3160450103. |
[23] |
N. Winter,
$W^{2, p}$ and $W^{1, p}$-estimates at the boundary for solutions of fully nonlinear, uniformly elliptic equations, Z. Anal. Anwend., 28 (2009), 129-164.
doi: 10.4171/ZAA/1377. |
[24] |
J. Zhang and S. Zheng,
Lorentz estimates for asymptotically regular fully nonlinear parabolic equations, Math. Nachr., 291 (2018), 996-1008.
doi: 10.1002/mana.201600497. |
[25] |
J. Zhang, M. Cai and S. Zheng,
Weighted Lorentz estimate for asymptotically regular parabolic equations of p(x, t)-Laplacian type, Nonlinear Anal., 180 (2019), 225-235.
doi: 10.1016/j.na.2018.10.013. |
[1] |
Tsung-Fang Wu. Multiplicity of positive solutions for a semilinear elliptic equation in $R_+^N$ with nonlinear boundary condition. Communications on Pure and Applied Analysis, 2010, 9 (6) : 1675-1696. doi: 10.3934/cpaa.2010.9.1675 |
[2] |
K. T. Joseph, Manas R. Sahoo. Vanishing viscosity approach to a system of conservation laws admitting $\delta''$ waves. Communications on Pure and Applied Analysis, 2013, 12 (5) : 2091-2118. doi: 10.3934/cpaa.2013.12.2091 |
[3] |
Kazuhiro Ishige, Ryuichi Sato. Heat equation with a nonlinear boundary condition and uniformly local $L^r$ spaces. Discrete and Continuous Dynamical Systems, 2016, 36 (5) : 2627-2652. doi: 10.3934/dcds.2016.36.2627 |
[4] |
Jong-Shenq Guo. Blow-up behavior for a quasilinear parabolic equation with nonlinear boundary condition. Discrete and Continuous Dynamical Systems, 2007, 18 (1) : 71-84. doi: 10.3934/dcds.2007.18.71 |
[5] |
R.G. Duran, J.I. Etcheverry, J.D. Rossi. Numerical approximation of a parabolic problem with a nonlinear boundary condition. Discrete and Continuous Dynamical Systems, 1998, 4 (3) : 497-506. doi: 10.3934/dcds.1998.4.497 |
[6] |
Mahamadi Warma. Parabolic and elliptic problems with general Wentzell boundary condition on Lipschitz domains. Communications on Pure and Applied Analysis, 2013, 12 (5) : 1881-1905. doi: 10.3934/cpaa.2013.12.1881 |
[7] |
Cong He, Hongjun Yu. Large time behavior of the solution to the Landau Equation with specular reflective boundary condition. Kinetic and Related Models, 2013, 6 (3) : 601-623. doi: 10.3934/krm.2013.6.601 |
[8] |
Alain Hertzog, Antoine Mondoloni. Existence of a weak solution for a quasilinear wave equation with boundary condition. Communications on Pure and Applied Analysis, 2002, 1 (2) : 191-219. doi: 10.3934/cpaa.2002.1.191 |
[9] |
G. Acosta, Julián Fernández Bonder, P. Groisman, J.D. Rossi. Numerical approximation of a parabolic problem with a nonlinear boundary condition in several space dimensions. Discrete and Continuous Dynamical Systems - B, 2002, 2 (2) : 279-294. doi: 10.3934/dcdsb.2002.2.279 |
[10] |
Patrick Winkert. Multiplicity results for a class of elliptic problems with nonlinear boundary condition. Communications on Pure and Applied Analysis, 2013, 12 (2) : 785-802. doi: 10.3934/cpaa.2013.12.785 |
[11] |
Khadijah Sharaf. A perturbation result for a critical elliptic equation with zero Dirichlet boundary condition. Discrete and Continuous Dynamical Systems, 2017, 37 (3) : 1691-1706. doi: 10.3934/dcds.2017070 |
[12] |
Hongwei Zhang, Qingying Hu. Asymptotic behavior and nonexistence of wave equation with nonlinear boundary condition. Communications on Pure and Applied Analysis, 2005, 4 (4) : 861-869. doi: 10.3934/cpaa.2005.4.861 |
[13] |
Marek Fila, Kazuhiro Ishige, Tatsuki Kawakami. Convergence to the Poisson kernel for the Laplace equation with a nonlinear dynamical boundary condition. Communications on Pure and Applied Analysis, 2012, 11 (3) : 1285-1301. doi: 10.3934/cpaa.2012.11.1285 |
[14] |
Mariane Bourgoing. Viscosity solutions of fully nonlinear second order parabolic equations with $L^1$ dependence in time and Neumann boundary conditions. Discrete and Continuous Dynamical Systems, 2008, 21 (3) : 763-800. doi: 10.3934/dcds.2008.21.763 |
[15] |
Jing Wang, Lining Tong. Vanishing viscosity limit of 1d quasilinear parabolic equation with multiple boundary layers. Communications on Pure and Applied Analysis, 2019, 18 (2) : 887-910. doi: 10.3934/cpaa.2019043 |
[16] |
Sami Aouaoui, Rahma Jlel. On some elliptic equation in the whole euclidean space $ \mathbb{R}^2 $ with nonlinearities having new exponential growth condition. Communications on Pure and Applied Analysis, 2020, 19 (10) : 4771-4796. doi: 10.3934/cpaa.2020211 |
[17] |
Mingyou Zhang, Qingsong Zhao, Yu Liu, Wenke Li. Finite time blow-up and global existence of solutions for semilinear parabolic equations with nonlinear dynamical boundary condition. Electronic Research Archive, 2020, 28 (1) : 369-381. doi: 10.3934/era.2020021 |
[18] |
Francesca Da Lio. Remarks on the strong maximum principle for viscosity solutions to fully nonlinear parabolic equations. Communications on Pure and Applied Analysis, 2004, 3 (3) : 395-415. doi: 10.3934/cpaa.2004.3.395 |
[19] |
Jesús Ildefonso Díaz, L. Tello. On a climate model with a dynamic nonlinear diffusive boundary condition. Discrete and Continuous Dynamical Systems - S, 2008, 1 (2) : 253-262. doi: 10.3934/dcdss.2008.1.253 |
[20] |
Alexander Gladkov. Blow-up problem for semilinear heat equation with nonlinear nonlocal Neumann boundary condition. Communications on Pure and Applied Analysis, 2017, 16 (6) : 2053-2068. doi: 10.3934/cpaa.2017101 |
2020 Impact Factor: 2.425
Tools
Metrics
Other articles
by authors
[Back to Top]